Calculation

“ Large.—With calculation large, one will be distinguished among his acquaintances for his skill in arithmetical calculations. He will be enabled to tell at a glance, operations which to an ordinary accountant require the use of figures. If Causality and Comparison are large, he will excel in solving difficult problems in the higher mathematics, but if these organs are deficient, his talent will be limited to arithmetical calculations. ”

“ To raise the number to ä higher power means in point of form to go on multiplying a number with itself an indefinite amount of times.—Since this third type of calculation exhibits the complete equality of the sole existing distinction in number, viz. the distinction between Sum or amount and Unity, there can be no more than these three modes of calculation. ”

“ In this point of view the numerical solution of questions which we cannot resolve algebraically, and even the calculation of "Definite Integrals," whose general integrals we do not know, really make a part, in spite of all appearances, of the domain of arithmetic, in which we must necessarily comprise all that which has for its object the determination of the values of functions. ”

“ Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization, just as the child of to-day learns the application of these laws by empirical methods. ”

“ To sum up, the aim of mathematical physics is not only to facilitate for the physicist the numerical calculation of certain constants or the integration of certain differential equations. It is besides, it is above all, to reveal to him the hidden harmony of things in making him see them in a new way. ”

“ It is also convenient to regard as coming under mensuration the consideration of certain derived magnitudes, such as the moment of a plane figure with regard to a straight line in its plane, the calculation of which involves formulae which are closely related to formulae for determining areas and volumes. ”

“ In the first section the formulæ were so combined as to render them subservient to the purposes of numerical calculation; in the second, these same formulæ were calculated for values of the variable, selected at certain successive distances; and under the third section, comprising about eighty individuals, who were most of them only acquainted with the two first rules of arithmetic, the values which were intermediate to those calculated by the second section were interpolated by means of simple additions and subtractions. ”

“ Therefore all these millionary equations of fives must have been produced by a cause, or causes, possessed of reason, or by a power destitute of that attribute. If we assume the first alternative there will be no chances for calculation, the efficient itself being amply adequate to develop the mathematical harmony. ”

“ We may even say, with perfect truth, so as to indicate in a word the general range of the science, that if we did not fear to multiply calculations unnecessarily, and if we had not, in consequence, to reserve them for the determination of the quantities which could not be measured directly, the determination of all the magnitudes susceptible of precise estimation, which the various orders of phenomena can offer us, could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles. ”

“ Calculation is not an experiment.
6.234 Mathematics is a method of logic.
6.2341 The essential of mathematical method is working with equations. On this method depends the fact that every proposition of mathematics must be self-evident.
66.24 The method by which mathematics arrives at its equations is the method of substitution.
For equations express the substitutability of two expressions, and we proceed from a number of equations to new equations, replacing expressions by others in accordance with the equations. ”

“ Some of the most conspicuously beautiful experiments in the whole range of science, have been devised for the purpose of indirectly measuring quantities, which in their extreme greatness or smallness surpass the powers of sense. All that we need to do, is to discover some other conveniently measurable phenomenon, which is related in a known ratio or according to a known law, however complicated, with that to be measured. Having once obtained experimental data, there is no further difficulty beyond that of arithmetic or algebraic calculation. ”

“ By means of this principle calculations can by made by persons whose business it is to do so, and stored up apart for use. The vast saving to mental labour by this simple and beautiful adjustment of numbers may be estimated by a glance at any collection of tables of logarithms. In a science like astronomy, progress would be terribly impeded if calculations had to be conducted by the ordinary methods. ”

“ In some other works which I have recently examined, I find a confusing mass of the most intricate mathematical calculations, abounding in an almost infinite number of characters, and extending over hundreds of pages, but on a close examination of some of the deductions arrived at, I find that a good many of the mathematical equations are based on a mistaken hypothesis, and the results arrived at are very wide of the truth. ”

“ I may here refer to a curious mathematical calculation by Dr. Thomas Young, to the effect, that if three words coincide in two different languages, it is ten to one they must be derived in both cases from some parent language, or introduced in some other manner. “Six words would give more,” he says, “than seventeen hundred to one, and eight near 100,000, so that in these cases the evidence would be little short of absolute certainty.” ”

“ Mathematics of the higher order expressed in elaborate formulæ do very well in communications between college professors—that is, if they happen to be agreed. When, however, these calculations are so intricate as to require a clever mathematician a whole day to study out the meaning of a single page, and if when the riddle is solved, we find that these calculations are based on a fallacy, and the results in conflict with facts, it becomes quite evident to the actual experimenter that they are of little value. ”

“ When we employ the multiplication-table in multiplying numbers of several places, and so use the results of old operations of counting instead of performing the whole of each operation anew; when we consult our table of logarithms, replacing and saving thus new calculations by old ones already performed; when we employ determinants instead of always beginning afresh the solution of a system of equations ”

“ It should be noted that the mathematical calculations provide data which are really only approximations, and consequently it is often found that a system constructed on such data requires modification before it fulfils the practical requirements. ”

“ The labor of the calculations and reductions in terms of weights and measures is reduced to a minimum by the simple relation between the units of mass and dimension and by the use of decimal parts and decimal notation. ”

“ Those calculations, as you probably know, if worked out without the aid of what, for a better term, I will call a mathematical ready reckoner, would take years to accomplish, and it was with the object of measuring the immeasurable, that Newton and Leibnitz produced their calculi. ”

“ Those who are unacquainted with the details of scientific investigation have no idea of the amount of labor expended in the determination of those numbers on which important calculations or inferences depend. They have no idea of the patience shown by a Berzelius in determining atomic weights; by a Regnault in determining coefficients of expansion; or by a Joule in determining the mechanical equivalent of heat. ”

“ It is composed of what is called the Calculus, [2] taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis. The Calculus has the solution of all questions [Pg 34] relating to numbers for its peculiar object. Its starting point is, constantly and necessarily, the knowledge of the precise relations, i.e., of the equations, between the different magnitudes which are simultaneously considered; that which is, on the contrary, the stopping point of concrete mathematics. ”

“ The fundamental problem of the Differential Calculus is to establish a measure of this change in the function with mathematical precision. It was while investigating problems of this sort, dealing with continuously varying quantities, that Newton [1] was led to the discovery of the fundamental principles of the Calculus, the most scientific and powerful tool of the modern mathematician.
26. Increments. The increment of a variable in changing from one numerical value to another is the difference found by subtracting the first value from the second. ”

“ We are almost constantly obliged, in physical science, to measure the quantities with which we deal. We measure a quantity when we compare it with some standard of the same kind. A simple number expresses the result of the comparison.
If we adopt arbitrary units of length, time, and mass (or quantity of matter) , we can express the measure of all other quantities in terms of these so-called fundamental units. A unit of any other quantity, thus expressed, is called a derived unit. ”

“ Mathematics contain properly the doctrine of measure; and the object of this science is commonly said to be quantity; therefore quantity ought to be defined, what may be measured. Those who have defined quantity to be whatever is capable of more or less, have given too wide a notion of it, which, I apprehend, has led some persons to apply mathematical reasoning to subjects that do not admit of it. ”

“ As science has been developed, the domain of quantity has everywhere encroached on that of quality, till the process of scientific inquiry seems to have become simply the measurement and registration of quantities, combined with a mathematical discussion of the numbers thus obtained. It is this scientific method of directing our attention to those features of phenomena which may be regarded as quantities which brings physical research under the influence of mathematical reasoning. ”

“ A measure, then, is a divisor which leaves no remainder.
92. When one number is a measure of two others, it is called a common measure of the two. Thus, 15 is a common measure of 180 and 75. Two numbers may have several common measures. For example, 360 and 168 have the common measures 2, 3, 4, 6, 24, and several others. Now, this question maybe asked: Of all the common measures of 360 and 168, which is the greatest? The answer to this question is derived from a rule of arithmetic, called the rule for finding the greatest common measure, which we proceed to consider. ”

“ The Calculus of Values, or Arithmetic, would appear, at first view, to present a field as vast as that of algebra, since it would seem to admit as many distinct questions as we can conceive different algebraic formulas whose values are to be determined. ”

“ In this, and in all other instances, as was explained above, the particular numerical data and the numerical results are determined by means and by portions of the mechanism which act quite independently of those that regulate the operations. In studying the action of the Analytical Engine, we find that the peculiar and independent nature of the considerations which in all mathematical analysis belong to operations, as distinguished from the objects operated upon and from the results of the operations performed upon those objects, is very strikingly defined and separated. ”

“ The mathematical questions which have given birth to the Calculus of Variations consist generally in the investigation of the maxima and minima of certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. ”

“ I shall now teach you how to multiply the unknown numbers, that is to say, the roots, one by the other, if they stand alone, or if numbers are added to them, or if numbers are subtracted from them, or if they are subtracted from numbers; also how to add them one to the other, or how to subtract one from the other.
Whenever one number is to be multiplied by another, the one must be repeated as many times as the other contains units. ”

“ The measures which are used in this country are not those which would have been chosen had they been made all at one time, and by a people well acquainted with arithmetic and natural philosophy. We proceed to shew how the results of the latter science are made useful in our system of measures. ”

“ The confusion, the difficulties, the contradictions which, in consequence of a want of accurate distinctions in this particular, have up to even a recent period encumbered mathematics in all those branches involving the consideration of negative and impossible quantities, will at once occur to the reader who is at all versed in this science, and would alone suffice to justify dwelling somewhat on the point, in connexion with any subject so peculiarly fitted to give forcible illustration of it, as the Analytical Engine. ”

“ They merely determine the operations [1] to be effected, which operations may of course be performed on an infinite variety of particular numerical values, and do not bring out any definite numerical results unless the numerical data of the problem have been impressed on the requisite portions of the train of mechanism. In the above example, the first essential step towards an arithmetical result, would be the substitution of specific numbers for , and for the other primitive quantities which enter into the function. ”

“ Such is the character of the judgments which the calculus enables us to form with respect to the occurrence of a certain difference between the real value of any quantity under measurement and the value assigned to it by the measurement. The confidence that the constants which we have determined are accurate within certain limits is a subjective feeling which cannot be dislodged from an important part of probabilities. ”

“ All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry the child's mind along with them in learning arithmetic; all who wish to teach numbers, and not mere ciphers—now teach it through the evidence of the senses, in the manner we have described.
We may, if we please, call the proposition, "Three is two and one," a definition of the number three, and assert that arithmetic, as it has been asserted that geometry, is a science founded on definitions. ”